Doing the Homework

So how do you actually solve homework problems? It may help to know how the mind solves problems.

You may be used to problems that you can solve at once, and you may imagine that all problems are solved this way: if you can't do a problem at once, you cannot do it at all. Problems in standardized exams are like this: a typical standardized exam problem is a simple test of one thing. But there are other kinds of problems, problems that are composed of several different pieces. To solve such a problem, you have to analyze it ("analyze" means literally, to take apart), and that takes time, energy, and sometimes creativity.

One difficulty is that the text should be a model, and the text contains such smooth and polished solutions and proofs -- compared to what students can produce -- that students may wonder if there is something special about the mathematicians who originally proved the theorems. But in fact, the mathematicians who originally developed the theorems did not come up with anything looking like what is in the textbook. For example ...

• First, there are some obscure results that seem to talk about esoteric things, like the assortment of results about lengths of lines and centers of mass by Gilles Personne de Roberval, Gregory St. Vincent, Evangelista Torricelli that we now say, with hindsight, led to the Fundamental Theorem of the Calculus. Frequently, it isn't clear at the time where (if anywhere) these results are going.

• Then comes the one or two or more mathematicians who come up with something that we call the Fundamental Theorem. James Gregory didn't seem to know the significance of his result, which is similar to that of Isaac Barrow, who did. Of course, this theorem seems very strange to us today, since it is constructed out of classical geometric notions.

• Then comes the one or more mathematicians who the popular books will say made the great accomplishment. For example, Isaac Newton (who was very conscious of his public image and cultivated a reputation for "genius") developed a system of mathematical results, including a version of the Fundamental Theorem of the Calculus; Newton, who was Barrow's student, is generally regarded as the inventer of the calculus.

• Frequently, the great accomplishment doesn't work right, or is unintelligible, or has errors, or, like Newton's Calculus, all of the above. Sometimes we need people, like Gottfried Liebniz, simply to make sense out of the invention. In Liebniz's case, he had to figure out what a function is (this after Newton was integrating and differentiating functions -- or at least that is what Newton said he was doing when he was dividing zero by zero(!?)).

• Okay, so people understand what the invention is about, but one is not supposed to divide zero by zero, so now comes the long train of people who work out ways to finesse the mess ( André Marie Ampère, who came up with one of the major early versions of the Mean Value Theorem), to clean up the mess ( Augustin Louis Cauchy, and his limits, although some limit-like things were already around), and even clean up the mess made by people who cleaned up the mess ( Karl Theodor Wilhelm Weierstrass, whose epsilons and deltas made the notion of the limit more precise but --- more work ahead, folks! --- less intelligible), which can lead some people to think that there must be a better way ( Abraham Robinson who used mathematical logic to circumvent all those nasty limits ... but at what cost?).

• And of course, then there are the people who write the texts: in the case of the Calculus, starting with Jean Le Rond d'Alembert, and his Cours d'Analyse. But that is not it, for the first text really ... well, we can and should do better. So after two centuries of successive textbook writers figuring out how to do things better than their competitors, we have the highly staged productions you can now buy for outrageous prices.

• Meanwhile, the research never ends, as mathematicians develop increasingly powerful versions of the Fundamental Theorem, from Green's Theorem to Stokes' Theorem to the recent index theorem of Michael Atiyah and Isadore Singer, for which they won the 2004 Abel Prize, the highest award in mathematics.

• Albert Einstein spent seven years developing his theory of General Relativity. This meant working on several interconnected complicated problems.

• Andrew Wiles spent seven years constructing a proof of Fermat's Last Theorem, a deceptively simple-looking problem (show that there is no tuple of four positive integers x, y, z, n, with n > 2, such that xⁿ + yⁿ = zⁿ).

To analyze a complicated or otherwise hard problem, one tries to take it apart, concentrate on it (or parts of it, or problems similar to it), return to it (because you probably won't get it all at once), explore it from a variety of perspectives, etc. That is what one should do consciously, with the goal of getting the Unconscious, the great machine in the largely unseen depths of your mind, to do some work. For it is the Unconscious that solves hard problems. To get your Unconscious to solve a problem, you must do the following:

• Keep badgering the Unconscious by returning to the problem and spending time on it. The Unconscious is somewhat lazy, and gauges how important things are by the amount of time and energy the Conscious invests on it: the way to get the Unconscious to work on something is to repeatedly work on it Consciously.

• Work on the correct problem. There is a difference between working on a problem and worrying about it. When working on the problem, one is conscious of the problem, and aspects and approaches to the problem, and if that is what one concentrates on, that is what the Unconscious will try to deal with. But if one spends the time worrying about the problem, obsessing over one's inability to solve it quickly, wondering what will happen if the problem doesn't get solved, all that will do is get the Unconscious to think up dire consequences of not solving the problem --- and thinking up dire consequences is one of those things that the Unconscious seems to like to do. Getting your Unconscious to work for you rather than against you requires being in the right state of Consciousness, and having the right attitude. So work on the problem, but avoid worrying about it.

• Try approaching it from various angles. There is not necessarily one right way to do it, so think up examples, find analogies and related problems, try to find different ways of looking at it, and in general, find additional material for the Unconscious to use.

So here is the procedure. Most homework problems are straightforard: here is a description of specifics about doing a mathematics assignment.

• First, if the problem is a hard one, you need to get some idea of what to do. This often involves brainstorming, a technique based on the principle of association.
• Then after a certain amount of analysis and synthesis, you need to construct the solution.
• Then you need to write it up.